Fixed point theorems for arc-preserving mappings of uniquely arcwise-connected continua

Fugate, J. B.; Mohler, Lee

doi:10.1090/S0002-9939-1995-1307517-4

Fixed point theorems for arc-preserving mappings of uniquely arcwise-connected continua
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by J. B. Fugate and Lee Mohler PDF

Proc. Amer. Math. Soc. 123 (1995), 3225-3231 Request permission

Abstract:

Let X be a uniquely arcwise-connected continuum and $f:X \to X$ a map which is arc-preserving (i.e., the image of each arc is an arc or a point). We prove that f has a fixed point.

References

K. Borsuk, A theorem on fixed points, Bull. Acad. Polon. Sci. Cl. III. 2 (1954), 17–20. MR 0064393
Charles L. Hagopian, Uniquely arcwise connected plane continua have the fixed-point property, Trans. Amer. Math. Soc. 248 (1979), no. 1, 85–104. MR 521694, DOI 10.1090/S0002-9947-1979-0521694-X
K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
A. Lelek and L. Mohler, On the topology of curves. III, Fund. Math. 71 (1971), no. 2, 147–160. MR 296911, DOI 10.4064/fm-71-2-147-160
Lee Mohler, The fixed point property for homeomorphisms of $1$-arcwise connected continua, Proc. Amer. Math. Soc. 52 (1975), 451–456. MR 391064, DOI 10.1090/S0002-9939-1975-0391064-8
Lee Mohler and Lex G. Oversteegen, Open and monotone fixed point free maps on uniquely arcwise connected continua, Proc. Amer. Math. Soc. 95 (1985), no. 3, 476–482. MR 806091, DOI 10.1090/S0002-9939-1985-0806091-X
Mirosław Sobolewski, A uniquely arcwise connected continuum without the fixed point property, Bull. Polish Acad. Sci. Math. 34 (1986), no. 5-6, 307–313 (English, with Russian summary). MR 874874
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108, DOI 10.1007/978-1-4612-5775-2
L. E. Ward Jr., A fixed point theorem for chained spaces, Pacific J. Math. 9 (1959), 1273–1278. MR 108784, DOI 10.2140/pjm.1959.9.1273
G. S. Young, Fixed-point theorems for arcwise connected continua, Proc. Amer. Math. Soc. 11 (1960), 880–884. MR 117711, DOI 10.1090/S0002-9939-1960-0117711-2